CONTINUED DEVELOPMENT OF NODAL METHODS
FOR REACTOR ANALYSIS
by
A.F. Henry, O.A. Adekugbe, W.H. Francis,
I.S. Muhtaseb, A.C. Onyemaechi,
T.A. Taiwo, E. Tanker
Energy Laboratory Report No. MIT-EL-85-003
March 1985
inn
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i
CONTINUED DEVELOPMENT OF NODAL METHODS
FOR REACTOR ANALYSIS
by
A. F. Henry
0. A. Adekugbe
W. H. Francis
I. S. Muhtaseb
A. C. Onyemaechi
T. A. Taiwo
E. Tanker
Energy Laboratory
and
Department of Nuclear Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Sponsored by
Consolidated Edison Company of New York
Northeast Utilities Service Company
Pacific Gas & Electric Company
PSE&G Research Corporation
under the
M.I.T. Energy Laboratory Electric Utility Program
Final Report for the Period: January 1, 1984 - March 31, 1985
M.I.T. Energy Laboratory Report No. MIT-EL-85-003
March 1985
101,
TABLE OF CONTENTS
Page
No.
I.
Introduction ..................................
1
1.1
Review of Earlier Work .........................
1
1.2
Summary of Present Contract Work ...............
5
1.3
Future Work ...................................
7
Studies Carried Out During the Present Contract
Period ..........
..........................
9
Comparison of the Koebke and Generalized Equivalence Approach to Determining Discontinuity
Factors ..................... ....................
9
II.
2.1
2.2
2.3
2.4
2.5
2.6
Determination of Approximate Discontinuity
Factors for BWR's ................................
10
Matching Transport Theory Results with Finite
Difference Diffusion Theory Models ............
11
The Use of Node-Averaged Discontinuity Factors
for Assembly-Sized Nodes .......................
25
Derivation of Simpler Nodal Models from the
QUANDRY Equations ..............................
26
Nodal Methods for Transient Analysis ...........
28
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I
Final Report on Continued Development
of Nodal Methods for Reactor Analysis
I.
Introduction
This report is a final summary of work carried out
with the support of Consolidated Edison Co., Northeast
Utilities Service Co., Pacific Gas and Electric Co., and
Public Service and Gas Research Corp.
The project was ini-
tially for the period January 1984 through December 1984,
but a no cost extension stretched its duration through
March 1985.
Work performed under the project has been concerned
with the development of Nodal Methods for Reactor Analysis.
The two-group, three dimensional, time-dependent nodal code
QUANDRY (1
)
has been the basis for the development.
Studies
carried out have been aimed at testing and increasing the
efficiency, capability, accuracy and reliability of this
code.
In addition, the code has been used to perform numer-
ical tests of certain standard design procedures for which,
hitherto, reference calculations have been too expensive.
In order to put the work, which will be summarized in
the following sections, into proper context, this introductory
portion of the report will first review older work, then summarize the results to be discussed in more detail in later
sections, and finally point out areas where further study
might be very fruitful.
1)
Review of Earlier Work.
Historically, the use of QUANDRY for reactor analysis
has evolved in three stages.
-1-
The original code (1
was aimed
-2-
at solvinq the static and time-dependent, two-group diffusion
equations for reactors composed of large homogeneous nodes.
It was found that, given the homogenized nodal parameters,
very accurate values of node-averaged powers (maximum error
% 2%) could be found.
No adjustment of albedoes or correc-
tion parameters to reference calculations was necessary.
Computer running times were orders of magnitude less than
those required by finite difference methods to obtain the
same accuracy.(1,2)
The second stage of evolution involved the development of systematic and accurate methods for determining the
homogenized group-constants needed by the code.
Here the
use of "discontinuity factors" (3,4) came into the picture.
These factors are a variant on the "heteroaeneitv factors"
first introduced by K6ebke.(5)
They correct the basic QUANDRY
equations so that reference results will be reproduced exactly.
Thus, if a quarter-core PDQ solution is available, edits from
that solution can be used to find discontinuity factors which,
when used in QUANDRY, will yield average nodal powers that
equal to within computer round-off error those edited from the
PDQ result.
Moreover, discontinuity factors found from local
assembly or color set calculations do almost as well (% 1%
maximum error in node-averaged power).
The final stage of QUANDRY development dealt with the
reconstruction of local fuel-pin powers -- the "dehomogenization" problem.
The approach here was to express the detailed
Mmi
-3-
heterogeneous flux shape throughout a node as the product
of a local "fine-structure" shape (found from a detailed
PDQ solution for an assembly, color set or quarter core at
the beginning of life) multiplied by a general quadratic
function of the coordinate directions.
The coefficients of
the quadratic -- twenty-seven of them in three dimensions --
are found by matching to the volume-averaged and faceaveraged fluxes of the node (which can be backed out of the
QUANDRY solution) and to the nodal corner point fluxes and
average fluxes along lines connecting corner points (which
can be obtained from the QUANDRY solution by interpolation.(6,7)
The reconstruction methods were tested through the beginning of the third depletion cycle for small, twodimensional PWR and BWR benchmarks, composed of realistic
fuel assemblies and surrounded by water reflectors including a stainless steel baffle for the PWR case).
For the PWR
case, with discontinuity factors and fine-structure flux
shapes found from assembly calculations for interior nodes
and from color sets extending into the reflector for peripheral nodes, maximum errors in reconstructed local fuel-pin
power were % 5% ( 2% for the highest power pins.89)
However, to obtain comparable accuracy for BWR's, it was
necessary to use color sets of larger size (comprising several assemblies) or to use iterative response matrix techniques.(1 0 )
For control rod tips at axial mesh interfaces,
comparable accuracy was obtained for three-dimensional
-4-
benchmark test problems at BOL, the reference results being
3D PDQ's run for us by Northeast Utilities.(7,8,10,11)
Many schemes for improving the efficiency or capability
of QUANDRY have been investigated since the initial version
of the code became available.
For example, if two-group
albedoes at the core-reflector interface are available from
reference calculations, these may be collapsed to equivalent
one-group albedoes and used along with one-group homogenized
cross sections and discontinuity factors computed from twogroup, zero-current-boundary-condition assembly computations
in order to obtain parameters for a one-group QUANDRY(12)
which yields power shapes and critical eigenvalues very
close to two-group QUANDRY predictions.
On the other hand, we have not succeeded in using computations for local regions to compute consistently accurate
albedoes for reflectors.
It appears necessary either to per-
form some reference 2D QUANDRY calculation with the reflector
explicitly included or to impose the albedoes conditions several mean free paths away from the core, rather than at the
core-reflector interface.(13)
Once found, however, the albedoes
seem to be acceptably accurate for situations differing significantly from the reference case for which they were computed.
One development that has been quite successful has
been the resolution of the so-called "rod-cusping" problem.
A way has been devised for employing discontinuity factors
at axial interfaces to account for a control rod partially
IYI1
----
-5-
inserted in a node.
The procedure can be made entirely in-
ternal to the code so that no auxiliary calculations are
required.
It appears to be very accurate for both static
and transient problems.(14)
Finally, a method suggested by Kord Smith
(15) for
reducing the storage requirements of the code by a factor
of five has been successfully tested.
Also, vis-a-vis the
supposed complexity of the code, it should be noted that a
static version is now available on an IBM PC.(16)
2)
Summary of Present Contract Work
During the present report period we have looked at a
number of schemes for improving the accuracy and efficiency
of QUANDRY.
Thus we ran a small test problem to compare the
accuracy of the K6ebke
procedures.
parable
(5 )
vs. the Smith
(3 )
homogenization
The accuracy of both methods appears to be com-
(Section II-1).
An attempt to improve the efficiency of the response
matrix method for determining BWR discontinuity factors during depletion calculations was unsuccessful
(Section 11-2).
On the other hand, a first look at a method for going
directly from a spectrum code, such as CASMO, to QUANDRY
(avoid-
ing both assembly-sized and quarter core PDQ's) is very promising
(Section 11-3).
This work indicates that, for pin-cell-
sized mesh intervals, it is often accurate to replace the
face-dependent discontinuity factors
(which cause finite
difference equations based on cell-homogenized cross sections
-6-
to match exactly transport calculations for the heterogeneous
cell) by their average values.
We have shown (Section II-4)
that this use of average discontinuity factors can be extended to assembly computations performed by the analytical
nodal method embodied in QUANDRY.
We have also shown that,
if the use of node-averaged discontinuity factors is accurate for assembly-sized nodes analyzed using the coarsemesh-finite-difference option in QUANDRY, the standard oneand one-and-a-half group nodal models EPRI-NODL-P/B, FLARE,
PRESTO, SIMULATE can be derived systematically from the
basic QUANDRY equations
(Section II-5).
We had hoped to test the QUANDRY scheme for reconstructing local pin-cell power against a quarter-core PDQ
analysis of Indian Point-2.
An attempt to apply QUANDRY to
a similar SALEM-1 quarter-core PDQ result had been only partially successful
powers % 3.9%),
(1 7 )
(maximum difference in node-averaged
since the PDQ color set edits needed to de-
termine discontinuity factors had not been requested when
the original PDQ's were run.
Through a contract with EPRI
we have now developed that edit capability and are currently
engaged in analyzing the first depletion cycle for ZION-2.
Unfortunately, manpower limitations have prevented Con
Edison from assembling and sending us all the original color
set and quarter core data we would need to carry out the
Indian Point-2 analysis.
out the analysis.
We still hope eventually to carry
---
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---
N111W
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Iii
-7-
One major effort which has been successfully carried
out during the year is the development and preliminary testing of point-kinetics and quasi-static methods for analyzing
reactor transients using expressions for the kinetics parameters (reactivity, prompt neutron lifetime, etc.) derived
from the nodal equations.
Comparison of the results pre-
dicted by these approximations with 2D and 3D reference
transient calculations provided by QUANDRY for simple benchmark problems suggests that the quasi-static scheme is fairly
accurate, but more expensive than the full space-time reference
calculations.
The point kinetics calculations differ from the
reference calculations by amounts that for some cases are unexpectedly large (Section II-6).
3)
Future Work
The homogenized two-group cross sections and discontinuity factors needed as input for QUANDRY can be edited
from the same computations used to determine the one or oneand-a-half group parameters needed for conventional nodal
schemes.
The overall procedure is sketched in our progress
report for the period September 1982 - December 1983.(18)
At present three groups (S. Levy, Inc.; N.U.S., and Studsvik of
America) are creating production versions of the code, which
should make the job of preparing input much more automatic.
Under these circumstances, we believe that future work at
M.I.T. should be concerned with testing how to use the code
in the most accurate and efficient manner.
We still hope to
analyze Indian Point-2 and to extend our comparison against
-8-
the first cycle depletion of ZION-2 to the reconstruction of
pin-cell powers.
We need to compare our depletion methods
with those of the KWU group (who claim that one must account
for the flux tilt across assemblies in doing depletion problems).
Finally, for BWR's, we need to develop a more effi-
cient way to find homogenized cross sections and discontinuity factors and to reconstruct pin-cell powers from the nodal
results.
With quarter-core PDQ's no longer needed, the possible
gain in accuracy of going to four-group color sets may be
worth exploring.
(The color set results would be used to
edit two- or one-group homogenized nodal parameters for
global QUANDRY calculations.
An even more accurate -- and
much more efficient -- procedure would be to edit QUANDRY
input directly from spectrum codes, such as CASMO or CPM,
run for color sets.
All PDQ calculations would be thereby
avoided.
Finally, we feel that more exploration of the transient area is important.
For computations for which point
kinetics is adequate, we should learn how to solve the
QUANDRY adjoint flux equations so that an accurate determination of reactivity coefficients based on 3D static nodal
computations will be possible.
In addition, with tests of
point kinetics calculations showing serious inaccuracies
and three-dimensional nodal calculations moderately expensive,
we should develop systematic ways of reducing three-dimensional
transient calculations to equivalent one-dimensional or "supernode" (100 nodes for a 3D model of the entire reactor) computations.
UI
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-9-
II.
Studies Carried Out During the Present Contract Period
1)
Comparison of the K6ebke and Generalized Equivalence
Approach to Determining Discontinuity Factors Oluwole A. Adekugbe.
In the original work of K6ebke for finding nodal equations that reproduce reference results exactly (Ref. 5) the
two extra degrees of freedom required for each group and each
direction in order to match reference results were obtained
by adjusting the diffusion coefficient and by imposing a
single discontinuity factor across the two inner faces of a
node in a given direction.
Thus for each group and each node
there are three different D g 's and three different discontinuity factors,
Generalized Equivalence Theory, as formulated by Smith
(Ref. 3),
introduces instead a single (arbitrary) value of Dg
for all directions and two different discontinuity factors for
each group and each direction.
Both schemes will reproduce exactly reference results
if "exact" discontinuity factors and/or directional D 's are
used.
The question arises, however, whether one method is
preferable when approximate correction parameters are used.
We have carried out a simple numerical test to examine this question for a small, one-dimensional reactor composed of six heterogeneous assemblies consisting of fuel slabs,
control slabs and water channels.
Two-group reference fluxes
were found for the reactor, and flux-weighted parameters
along with exact Ko6bke and Smith parameters were found.
The
-10-
reactor was then perturbed by changing cross sections and/or
external boundary conditions.
New, fine mesh reference cal-
culations were performed, and these were compared with nodal
calculations found using the unperturbed (and hence no longer
"exact") K6ebke and Smith parameters.
Resultant errors in
core eigenvalue are displayed in the following table:
% Error in Eigenvalue
Case
Smith Parameters
1
2
3
K6ebke Parameter
0.66
-0.27
-0.27
.24
-0.29
+0.59
These results suggest that there is little to choose
between the two methods.
Since the Kaebke scheme requires
iteration and can in some cases lead to indeterminate results,
we propose to continue using the generalized equivalence
(Smith) scheme.
2)
Determination of Approximate Discontinuity
Factors for BWR's - A. C. Onyemaechi.
References (18) and (19) show that, for PWR's, computing
discontinuity factors by running fine-mesh, zero-currentboundary-condition PDQ calculations for individual assemblies
or color sets yields discontinuity factors and homogenized
group-parameters which, when used in QUANDRY, reproduce reference values of node-averaged powers with a maximum error
of < 1.5%
However, to obtain comparable accuracy for BWR's,
an iterative procedure using response matrices for extended
assemblies appears to be required.
M
NMIIN
111011101411'
11-i
I AI
,L
j1WINIi
,
-11-
To avoid the significant cost of computing response
matrices by fine mesh finite difference methods, we have attempted to determine them using reconstructed, fine-mesh flux
shapes found by the (relatively cheap) methods used successfully for PWR's (and described in References 18 and 19).
Unfortunately, two different schemes applied to solve
the non-linear iterative equations which result have both
diverged.
Thus a cheap method for finding discontinuity
factors for BWR's that yield maximum errors in nodal power
of less than 2% has yet to be found.
3)
Matching Transport Theory Results with Finite
Differerence Diffusion Theory Models - Ediz Tanker
It is becoming standard practice for some utilities
to use sophisticated multigroup transport theory codes such
as CASMO or CPM applied to an entire fuel assembly in order
to obtain two-group diffusion theory parameters for fuel pin,
burnable poison pin and control rod cells.
The procedure for
doing this is generally to edit two-group cross sections for
the various pin cells from the multigroup, multiregion, transport theory results and then to adjust these values so that
fine mesh PDQ calculations run for the heterogeneous assembly
match the more important reaction rates (poison pin and fuel
cell absorption).
Part of this adjustment is needed to cor-
rect for mesh size effects, one mesh square per pin cell being too large a size to provide an accurate solution of the
two-group diffusion equations solved by finite difference
methods.
-12-
A more systematic and accurate procedure for finding
the equivalent diffusion theory parameters is to take advantage of the fact that, by using discontinuity factors it is
possible to derive coarse mesh finite difference (CMFD) equations that will reproduce reference results to within machine
round-off error.
&x+ +
2Dijk
-h I
f ik
-
rgf i+l,Jk i+l,jk
2
Di+l,jk
fijk+1
z
i,j,k+1 , j,k+1
fijk
k
2
fi-1ijk i-1,jk
gx+
Dijk
f ijk
f i,J-l,k
fi,j-1,k i,j-1,k
g
gY+
2DikJ
2D~ j-1,k
fijk
f ijk-1
gz+
,
2Dijk-1
-h
i
fijk 0ijk
gx- "g
g
-1
+
h
f ijk ijk
gz+ g
1 -1
x-
gx+
g
gz+
.g
g
Si-1,jk
g
-1
2 Dij,k+l
2Dijk
h8
gy+
g
gy-
2 Dij+lk
+±
+
2Di-
ijk
i
fjk
gx+ g
fij+1,k i, j+l
+ ...
f ijk
h
g
gx-
Ij+1, k - -1
2 Dijk
-h
1-1
fi+1,Jk
fiJk
- h
Such equations have the form
fijk
ijk
-1
,k-1
z+
zDijk
2
ij
'
k-1
-
ijk ijk
gz-
1
+ h3 ijk ijk
G
Sh 3
g
Mi jk ijk
l1,2...G;
+
h3
i=1,2.. . I-1;
ijk ijk
J
1,2... J-1;
(1)
k=1,2...K-1
I
--
,111111 mlh
-13-
where fijk
gu+ is the group-g discontinuity factor for the (+)
side of node (ijk) in the uth direction (u = x y or z) and
fijk is its value on the (-) side, and the mesh size, h,
guhas, for simplicity, been taken equal in all directions.
in Equation (1),
Zijk is the removal cross section
gg
from group-g' to group-g, and
ijk
ijk
ijk
gg
totg
g
=
ijk
gg
1
n E(ijk)n
g
fg
(2)
where the sum is over all fissionable isotopes, n.
If all the discontinuity factors in (1) are set to
unity, the conventional, "mesh centered" --
of the finite difference equations results.
not PDQ --
form
Moreover, the
same finite difference form results if the discontinuity
factors for a given group-g, and a given node, (ijk) are
replaced by their average values.
fi
gu±
Thus, if we have
= fijk ; u = x,y,z , defining
g
Dijk =
g
Dij
k
k
f13k
g
Aijk
^ijk
(ijk)g -
(a)g
ik
;(a) = tot, f or g'
g
Aijk
g
ijk ijk
g
g
(3)
-14-
yields the conventional finite difference equations.
Aijk
that the transformation (3) is such that Zijk
0
Note
ijk _
ijk
ijk
(a)g gjk . Thus all reaction rates are preserved.
In view of the possibility of correcting for transport effects by the very simple procedure of redefining
group parameters according to (3),
it is important to deter-
mine the error that results when the exact, face-dependent
fijk 's in (1) are replaced by average values fjk
gu
g
Also
it is important to determine the error that results when exact
parameters, determined for zero-current boundary conditions on
the faces of the assembly, are used when, in fact, the currents
are non-zero.
To shed light on these questions we have performed calculations for the PWR assembly shown in Figure (1).
Since
neither CASMO nor CPM is available to us, we used as reference calculations two-group, finite difference diffusion
theory solutions for the assembly of Fig. (1) with homogenized fuel cells (the white squares) and control rod cells
(the black squares) replaced by two-region cells composed
of a central square of pure fuel or control rod material surrounded by pure water, as is shown (for a quarter of the assembly) in Figure (2).
(If the control elements are removed,
the black squares in Fig. (1) and the shaded squares in
Fig. (2) become pure water.)
QUANDRY was used to solve Equas. (1) for the "superheterogeneous" reference cases.
(For these reference cases
WNIINiNi-i
Il
-15-
all f's were taken as unity.)
For a J
n = 0 boundary con-
dition on all faces QUANDRY then edited exact cell-averaged
D 's and E 's and face-dependent fgu's so that when Equas.
(1) were solved for the assembly of Fig. (1) with J *n = 0
boundary conditions, the reference values were again obtained.
The face-dependent and arithmetically averaged discontinuity factors for the JS-g- *n = 0 , boundary condition,
control-rod-free case are shown on Figure (3).
The figure shows that the exact, face-dependent discontinuity factors are all very nearly 1.000 for the fast
group.
For the thermal group they are in the range 1.000 -
1.008 for fuel cells and r' 1.025 for all four faces of the
water holes.
Thus, approximating them by their average
value is expected to yield accurate results.
That this is
indeed the case is shown by the results in Table (1).
% Error
in X
Reference
1.259204
0.0
Arithmetic
Averaged f's
1.258961
-0.02
Table 1.
Max. % Error
in Power Dist.
0.0
±0.40
Unrodded Assembly; Error in Eigenvalue and Power
Distribution Due to Use of Arithmetically AverageO
Discontinuity Factors for Mesh Cells (J*n = 0
Boundary Conditions)
For the very fine-mesh "super-heterogeneous" problems, it
was necessary to run QUANDRY in double precision and to
alter the conventional iteration procedure. Otherwise the
problem will not converge. Kord Smith made such runs for
US.
-16-
---- ---2 .. n
o
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21 ca
dimensions of each call
Fig. 1.
.
1.4 cm by 1.4 e
Heterogeneous PWR Assembly Geometry
-------
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-17-
Fig.
2.
2.
-ig.
A "Superheterogeneous"
O-W-
Model Showing Explicit
Explicit
Model
"Superheterogeneous"
A
(Square)
Fuel and Control
RodsShowing
0999
Cbas
1.004
e.ggl3 hfo.sg,
1.000
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Fig. 3.
Exact and Average Discontinuity Factors for
the Rod-Free Assembly
Alormhd
Pu0er
Auage &wcnd.
av
Fcd.
e Asco/. rdI.
m l
MIIIIm
N
-19-
The difference between the reference (superheterogeneous
geometry all f's = 1) and that using homogenized cell constants with exact face-dependent discontinuity factors calculated from the reference results is in the round-off
range and is not shown.
Tables (2) and (3) show the errors when reference,
face-dependent and averaged discontinuity factors, found for
the Jon = 0 reference problem, are used for cases where
J*n ; 0 .
For Table (2) a uniform albedo
boundary con-
dition was imposed for both energy groups on all four faces.
This condition was based on an estimate of the materials
buckling of the assembly.
Thus the eigenvalue for the as-
sembly with this boundary condition imposed is close to unity.
1
Reference
% error
in X
Max. % Error
in Power Dist.
1.003824
0.0
0.0
Reference
1.003921
0.01
-0.04
Arithmetic
Averaged f's
1.003739
-0.01
-0.40
Face-Dependent
f's from
J
*n = 0
Table 2.
Unrodded Assembly; Error in Eigenvalue and Power
Distribution Due to Use of Arithmetically Averaged
Discontinuity Factors for Mesh Cells (Uniform
Albedo Boundary Conditions Based on Estimate of
the Materials Buckling)
For Table (3) zero current boundary conditions were
imposed on the two inner faces of the quarter assembly, and
-20-
a boundary condition appropriate to a steel baffle followed
by a water reflector were imposed on the other two-faces.
% Error
Max. % Error
X
in X
in Power Dist.
0.4547454
0.0
0.0
Reference
0.4566759
0.42
1.30
Arithmetic
Averaged f's
0.4565254
0.39
1.24
Reference
Face-Dependent
f's from
J
*n = 0
- -
Table 3.
Unrodded Assembly; Error in Eigenvalue and Power
Distribution Due to Use of Arithmetically Averaged
Discontinuity Factors for Mesh Cells (J-n = 0
Boundary Conditions on Two Sides; Aibedo Boundary
Conditions Appropriate to a Baffle and Water ReFlector on Other Two Sides)
For these cases, use of the face-dependent discontinuity factors based on J -n = 0 boundary conditions on all four
-g faces is an approximation. Table (2), however, shows that for
uniform albedo boundary conditions it is an excellent one,
even though the eigenvalue has changed 25%.
For the simulated baffle-reflector boundary condition,
Table (3) shows an unacceptably large error in eigenvalue
when either the face-dependent or averaged discontinuity
factors based on the J .n = 0 reference assembly calculation
are used.
The perturbation in boundary conditions (which
caused the assembly eigenvalue to change from
0.455) is perhaps unrealistically extreme.
cause, more work is called for in this area.
'
1.25 to
Whatever the
--
U-h"
-21-
The face-dependent and averaged discontinuity factors
for the rodded assembly with J*n = 0 boundary conditions imposed are shown on Fig. (4).
Examination of these results
shows that use of an arithmetic average value of the discontinuity factors for mesh cells adjacent to a rodded cell is
unlikely to be a good approximation.
On the other hand, if
averaged discontinuity factors are used for all fuel cells
and for the inner side of the faces of a rodded cell, and
reference values are used for the outer sides of rodded cells
(i.e., for the inner sides of faces that adjacent cells have
in common with the rodded cell), discontinuity factors for the
other three faces of adjacent cells being replaced by the arithmetic averages, greater accuracy ought to result.
We shall call
this approximation "(3+1) discontinuity factors."
Table
(4) shows that greater accuracy does indeed result from use
of the (3+1) approximation.
Max. % Error
in Power Dist.
X
% Error
in X
Reference
0.8908857
0.0
Arithmetic
Average of
Discont.
Fact.
0.9060568
1.70
-3.47
(3+1) Discont.
0.8913067
Factors
0.05
-1.65
Table 4.
0.0
Rodded Assembly; Error in Eigenvalue and Power
Distribution Due to Use of Arithmetically Averaged and (3+1) Discontinuity Factors.
(J*n = 0 Boundary Conditions)
1939
0.995
1.ol
Ib.ol
0,9b
A.3
20.93-0.SS-
0.233
0.99
I.013
0 -993
I 9
03
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0. I.000
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6-199
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(
4. Exact and Average Discontinuity Factors
for the Rodded Assembly
o05 ri0.999
i 0. 9
o
mmlii.WNi
m
n
i.,1111
-23-
Tables (5) and (6) show the analogous results when
the albedo boundary conditions of Tables (2) and (3) are
used for the rodded case, the individual or averaged disJ .n
continuity factors being those from the -g
- = 0 roddedassembly calculation.
In Table (5) the effect of using
reference f's for the mesh cells adjacent to the rodded cell
is also shown, while in Table (6) the (inaccurate) results
from using averaged f's for all cells are omitted.
Rodded
A
% Error
in A
Max. % Error
in Power Dist.
1.003863
0.0
- Reference
1.003897
0.003
Arithmetic
Averaged f's
1.021008
1.71
3.47
4 Different f's
for Neighboring
Cells
1.004107
0.02
0.61
Reference
0.0
Face-Dependent
f's from
J
*n = 0
Table 5.
-0.04
Rodded Assembly; Errors in Eigenvalue and Power
Distribution Due to Use of Approximate Discontinuity Factors (Uniform Albedo Boundary Conditions
Based on Estimate of the Materials Buckling)
-24-
Rodded
Albedo from
EPRI-9
A
% Error
in A
Reference
0.7185753
0.0
0.0
Face-Dependent
f's from
J-g *n
- = 0
Reference
0.7187487
0.02
-0.05
(3+1) Discont.
Factors
0.7186226
0.007
-1.85
Table 6.
Max. % Error
in Power Dist.
Rodded Assembly; Errors in Eigenvalue and Power
Distribution Due to Use of Approximate Discontinuity Factors (J-n = 0 Boundary Conditions on Two
Sides; Albedo Boundary Conditions Appropriate to
a Baffle and Water Reflector on Other Two Sides)
For these rodded cases, the error in eigenvalue associated with using f's from a J -n = 0 reference assembly
calculation when the assembly is subject to much different
albedo conditions is much smaller (Table (6) vs. Table (3)).
There is some room for improvement in the power distribution
predicted by the (3+1) approximation, but that approximation
appears to be a vast improvement over using averaged discontinuity factors for all cells in the rodded assembly.
Many more details concerning the above results appear
in Reference (20).
These results are of course preliminary.
Nevertheless, we feel that this systematic approach for obtaining finite difference equations that will reproduce reaction and leakage rates predicted by spectrum codes is well
worth pursuing.
The eventual goal is to avoid completely
the need to perform any fine-mesh diffusion theory computations.
-WWNNNINIII
II All,
Yiili
lilIl Y Y
Y
IY II IIIIIIII 1,
11
"
11IIIY
ilidd
,,iiioomm
milli,
-25-
4)
The Use of Node-Averaged Discontinuity Factors
for Assembly-Sized Nodes - I. S. Muhteseb
The procedure of replacing face-dependent discontinuity factors by their average value can be extended to assemblysized nodes comprising an entire reactor.
The advantage is
a reduction in the number of input numbers.
Perhaps more
important is the fact that, if one can use average, rather
than face-dependent discontinuity factors for the nodes,
the transformation specified by Equation (3) will remove all
explicit f's from the QUANDRY equations.
Thus a code such
as TITAN, which at present has no provision for including
discontinuity factors, can be made more accurate.
We have tested the use of assembly-averaged discontinuity factors for the analytic nodal method with the quadratic transverse leakage approximation; i.e., for the regular form of the QUANDRY equation (often called Quad-QUANDRY
to distinguish it from the simple CMFD form).
The approxi-
mation has been applied to
i)
The EPRI-9 rodded benchmark -- a small PWR (see
Ref.
ii)
iii)
iv)
(18), Fig. 11)
The EPRI-9 unrodded benchmark
The Salem-I reactor
The HAFAS benchmark -- a small BWR (see Ref. 3)
In order to obtain acceptable accuracy it was found
necessary to use full assembly-sized nodes, which, except for
a few of the burnable-poison-loaded Salem assemblies, have
-26-
ninety degree rotational symmetry.
Also it was necessary
to represent reflector effects by an albedo boundary condition and to adjust the albedo so that the discontinuity
factor at a reflector interface of a peripheral assembly
could be arbitrarily set equal to the average of the other
three (almost equal) "internal" discontinuity factors.
(It
is legitimate to do this, since it is the quotient of the
albedo and the discontinuity factor that appears in the
nodal equations.)
QUANDRY results
Under these conditions the errors in
(due solely to replacing face-dependent
discontinuity factors by their averages) are given in
Table (7).
Table 7.
Error Due to Replacing Face-Dependent Discontinuity
Factors by Average Values in QUANDRY
Reac tor Model
(i)
(ii)
(iii)
(iv)
% Error in
Core Eigenvalue
Maximum % Error
in Nodal Power
0.01
0.01
-0.01
-0.04
-0.56
-0.10
-2.53
6.13
Except possibly for the HAFAS BWR model
proximation appears to be acceptable.
of this study is given in Ref.
5)
Average % Error
in Nodal Power
0.20
0.05
0.68
2.84
(iv),
the ap-
A complete description
(21).
Derivation of Simpler Nodal Models from the QUANDRY
Equations - Winston H.
It has already been shown
Francis.
(Section II-3 Eq. 1) that by
using discontinuity factors it is possible to derive coarse
mesh finite difference equations (CMFD) that will reproduce
---I ---- -
- __ _ -
-27-
reference values of node-averaged fluxes.
In Section II-3
the equations were used for pin-cell sized mesh intervals.
However, they remain valid even if the nodes are heterogeneous and assembly-sized.
Moreover, as pointed out in
Section 11-3, if the face-dependent discontinuity factors
for each node can be replaced by single, node-averaged
values, Equas. (1) reduce to a standard finite difference
form.
This situation raises the possibility of deriving
parameters for the standard nodal equations currently used
by utilities in a systematic, non-iterative fashion.
To this end we started with the two-group version of
Equa. (1) (with all f's = 1) and derived in a formally exact
manner the basic equations of SIMULATE, PRESTO and FLARE.
As
a result, it is possible to determine the albedos and adjustable parameters (the "g" of FLARE and the "a" of PRESTO, etc.)
directly from QUANDRY.
Not unexpectedly, the albedos and ad-
justable parameters turn out to be node-dependent.
This may,
however, be a price worth paying to avoid iterative searches
for albedos and fitting parameters and for greater reliability
and accuracy when these simple models are used.
We are having considerable trouble testing these reduced equations.
The first step of our testing procedure has
been an attempt to show that, with proper redefinition and
restriction of input, QUANDRY can be made to reproduce FLARE
results.
But the version of FLARE available to us seems to
-28-
treat albedos for corner assembly nodes in a fashion that
we have been unable to understand and to determine variations
in axial parameters in an automatic and internal manner involving thermal feedback, which precludes our computing the
"equivalent" input for QUANDRY problems.
Thus we cannot run
"equivalent" QUANDRY and FLARE problems except for very trivial
cases.
Our present disposition is to accept the (positive)
evidence of these very simple test cases that QUANDRY can
be made to solve the FLARE and PRESTO equations.
We can
then get on with the investigation of the accuracy of
eigenvalue and power shape computations determined using
FLARE and PRESTO adjustable parameters obtained systematically from higher order QUANDRY computations.
6)
Nodal Methods for Transient Analysis - T. A. Taiwo
Heretofore it has been virtually impossible to evaluate the numerical accuracy of point kinetics methods when
applied to transients involving three space dimensions.
One difficulty was that of obtaining an accurate threedimensional reference.
QUANDRY.
That was overcome with the advent of
A corresponding difficulty was that of computing
reactivity coefficients from a nodal solution.
The perturba-
tion formula could not be used because a code to solve the
adjoint equations was not available and because, even if
there had been such a code, there was no way to evaluate the
gradient terms 6D V*.V# from a nodal solution.
Thus the
~~Y
rr IIIII
III
~~~
IIIl
III1
1,111if
iiiiul III
JIM-
NIMYNIIIIMu
-29-
only practical ways to obtain reactivity coefficients were,
(1) to perturb temperatures and densities uniformly over the
entire core and perform three-dimensional nodal calculations
to find the associated changes in critical eigenvalue and
thence average reactivity coefficients for the entire core
(use of which leads to kinetic predictions of questionable
accuracy if temperature and density changes are nonuniform),
or (2) to perturb temperatures and densities in each node
and perform separate criticality calculations for each of
several hundred -- or even thousand -- perturbed cases
(a
very costly procedure).
We have overcome the difficulty of determining the
gradient term by deriving a perturbation formula based on
the QUANDRY matrix equations (rather than on the differential equations from which they are derived).
Unfortunately,
however, the formula requires that we know the solution of
the adjoint form of these matrix equations, and, although
it is simple to write down that adjoint form, we have not
as yet found a convergent solution method.
Despite this difficulty, we have been able to make
what we feel is a meaningful comparison between full spacedependent solutions to transients and corresponding solutions based on a consistent point kinetics model.
The trick
is to perform the space-dependent reference calculations using the CMFD form of the QUANDRY equations (the timedependent counterpart of Equa. (1)).
We next assume that
-30-
the discontinuity factors which cause the CMFD solution to
match exactly the full QUANDRY solution for the initial steady
state condition remain constant during the transient.
We then
find the adjoint of the steady state CMFD eauations.
(A code
to find adjoints for the CMFD form can be written fairly
easily.)
It is then possible to derive a perturbation ex-
pression for reactivity based on the CMFD equations and their
adjoints.
This formula requires that we obtain only two full-
core nodal solutions, the critical CMFD solution and its adjoint, for the steady state reference condition.
The essential assumption in this whole procedure is
that the discontinuity factors which make the reference CMFD
equations accurate do not change during the transients.
Our
attempts to validate this assumption indicate that it is
fairly accurate.
(2 3 )
However, we feel that the question of
its accuracy is not an important one for the purposes of comparing space-dependent and point kinetics solutions for the
same problem, provided we keep the discontinuity factors constant for both calculations.
We imposed that restriction for
all the test cases described in the following section.
For all numerical tests a two-group CMFD solution was
taken as the numerical standard.
Temperature changes in
each node were computed by the simple constant-pressure,
constant flow, no-boiling heat-transfer model embodied in
the original QUANDRY.(1)
These temperature changes lead to
changes in the two-group parameters of each node and thence
II_~ _
~_
-31-
to changes in core reactivity.
For the point kinetics cal-
culations, temperature coefficients for each individual node
were computed by using the initial, steady-state adjoint flux
for the CMFD model.
Point kinetics calculations based on
core-averaged temperature changes and full-core reactivity
coefficients were also made.
For three dimensional cases analyzed by point kinetics,
two methods for accounting for control rod motion were tested.
The first scheme was a totally consistent perturbation theory
method in which rod worths were computed by the perturbation
theory formula using initial flux and adjoint flux shapes.
For the second scheme, curves of reactivity vs. control rod
position were computed from a sequence of criticality calculations all run for the initial reactor temperature distribution.
Reactivity contributions from these curves for the
control rod at any particular position were added to changes
due to temperature feedback as computed by the perturbation
theory expression to obtain the total reactivity at particular times.
The Quasi-static method
(2 2 )
was also tested.
This
scheme is an improvement over point kinetics in that the flux
shape used to compute reactivity is up-dated from time to
time by freezing the temperature profile then present in the
core and doing a fixed-source static calculation.
The more
frequent the updates, the more accurate (but more expensive)
the quasi-static computation becomes.
-32-
Both two- and three-dimensional test cases have been
run for the various approximate methods.
These are all dis-
cussed (along with derivations underlying the theory) in
Reference (23).
Three particularly significant cases are
the following:
a)
EPRI-9 Control Rod Withdrawal; No Feedback
The EPRI-9 benchmark is shown as Figure (5).
It
models a small, PWR reflected by a one inch thick baffle,
followed by a water reflector.
are pure water.
Top and bottom reflectors
The transient is caused by removing the
control rod 12 cm. in 0.08 sec.
For feedback neglected,
results are displayed in Table (8).
The quasi-static results are fairly good, seven updates, however, being little better than six.
are more expensive.
Also, they
The consistent point kinetics calcu-
lation is extremely inaccurate.
Evidently computing the
rod worth with the unperturbed flux yields a total reactivity input less than prompt critical, whereas the reference
calculation corresponds to a super-prompt critical condition.
Use of the rod worth calibration curve significantly reduces
the problem, although the final power level is still underpredicted by 18%.
(b) EPRI-9:
Control Rod Withdrawal with Temperature
Feedback
Table (9) displays the comparisons of the various approximate kinetics model with the CMFD reference for the case
-33-
F-1
F-4
F-2.
F- I
F-4
#1 -3 :0
4.V-
F-2.
UVrojole0-
Se 4i
yz oc-de
LOFI
______A6edo
-4-o
Axial
Vie
WU
Fia. 5
EPRI-9:.
3D. Geometry
Sechln "
B.C.
TABLE 8
EPRI-9:
CONTROL ROD WITHDRAWAL PROBLEM WITHOUT FEEDBACK
NORMALIZED REACTOR POWER VS. TIME
Point Kinetics
(Rod Worth by
Perturbation
Theory
Quasi-static
6 Flux
Updates )
Quasi-static
0.000
1.000
1.000
1.000
1.000
0.025
1.274
(-1.0)
1.364 (-1.0)
1.380
( 0.1)
1.378
0.050
1.702 (-28.7)
2.345 (-1.8)
2.345 (-1.8)
2.381
(-0.2)
2.386
0.075
2.312
(-62.8)
6.179
(-0.6)
6. 179
(-0.6)
6.110
(-1.7)
6.215
1.000
2.742
(-86.7)
18.940
(-7.9)
19.960
(-3.0)
19.551
(-5.0)
20.569
0.125
2.810
(-94.2)
44.320
(-8.0)
48.648
( 1.0)
44.217
(-8.2)
48.172
0.150
2.828
(-97.2)
95.800
(-5.0)
105.098
( 4.2)
89.155
(-11.)
100.817
0.175
2.839
(-98.6)
197.249
(-2.0)
215.446
( 7.0)
171.193
(-14.)
201.211
0.200
2.849
(-99.2)
393.155
(-0.1)
428.852
( 9.0)
321.118
(-18.)
393.426
Time (s)
Comp. Time
(- 7.5)
(2.24)
1.364
(23.05)
7 Flux
Updates)
(22.47)
Point Kinetics
(Rod Worth
from Table
Reference
1.000
(1.86)
(5.5)
0
a
TABLF 9
EPRI-9:
CONTROL ROD WITHDRAWAL PROBLEM WITH FEEDBACK
NORMALIZED REACTOR POWER VS. TIME
Point Kinetics
(Rod Worth by
Perturbation
Theory
Time (s)
0.000
0.025
0.050
0.075
0.100
0.150
0.200
0.450
0.700
0.950
1.200
1.00000
1.22861
1.54203
1.93224
2.14362
2.16890
2.17424
2.17453
2.14508
2.10444
2.06127
Comp. Time
(-
7.0)
(-23.9)
(-49,8)
(-68.8)
(-77.2)
(-78.4)
(-71.9)
(-57.3)
(-44.0)
(-34.2)
(3.27)
Point Kinetics
(Rod Worth from
Table; Feedback
Quasi-static
(11 Flux Update)
1.00000
1.30785
1.98194
3.72440
6.67886
9.39926
9.96237
7.37212
5.01442
3.64861
3.15282
(26.36)
(-1.0)
(-2.2)
(-3.1)
(-2.9)
(-1.3)
(-1.3)
(-4.7)
(-0.2)
(-2.9)
( 0.6)
from Local
Temperatures)
1.00000
1.38003
2.38127
6.10970
19.54880
79.94714
125.3680
5.03650
2.71196
2.12771
1.91468
(2.86)
Point Kinetics
(Rod Worth from
Table; Feedback
from Coreaverage Temp.)
1.00000
( 4.5)
(17.5)
(5S.9)
(184.)
(739.)
(1141)
(-34.)
(-46.)
(-43.)
(-39.)
1.38099
2.38341
6.11635
19.47240
79.08880
150.24200
5.91509
2.99543
2.24880
1.94288
(
3.8)
(17.6)
(59.1)
(183.)
(730.)
(1 388)
(-23.)
(-40.)
(-40.)
(-38.)
Reference
1.00000
1.32072
2.02716
3.84550
6.87888
9.52692
10.09550
7.73183
5.02341
3.75662
3.13405
(6.67)
-36-
of feedback present.
Point kinetics with rod worth computed
by perturbation theory seem not to be so bad as for the nofeedback case.
But this is only because feedback has brought
the CMFD power back down again.
Again the quasi-static method
is acceptable, but almost four times as expensive as the CMFD
reference.
The point kinetics calculations based on pre-
calibrated control rod worths are both absurd, the computation using average temperature changes being the worse of
the two.
Evidently use of the unperturbed flux shape to
compute temperature feedback leads here to serious error.
(c) A Transient Induce by Changes in Coolant Inlet
Temperature
Table (10) shows the behavior predicted by various
models when the inlet coolant temperature is changed according to the formula
Tin = 533 - 37.3333t + 20.00t 2
The reactor model involved is similar to EPRI-9 except that the four corner positions have been made fuel assemblies, and a no-returning-current boundary condition has
been applied over the radial surface of the
(now square) core.
The point kinetics with reactivity feedback computed
as a sum of contributions from every node (Column 3) does
fairly well once the colder entering water has reached the
outlet of the axial zone (0 0.6 sec.).
On the other hand,
use of a core-averaged temperature coefficient yields results significantly different from the reference values.
With 18 flux updates the quasi-static method is fairly
e
a
TABLE 10
VARIABLE INLET TEMP. FLOW PROBLEM WITH FEEDBACK
= 533 - 37.3333T + 20.00T2 )
(Tc
NORMALIZED REACTOR POWER VS. TIME
Time (s)
0.0
0.1
0.2
0.3
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Reference
1.00000
1.26994
1.89087
3.05521
5.52793
11.09740
7.80197
5.40020
3.59296
2.34338
1.54674
Comp Time
(7.26)
Point Kinetics
(Feedback
from Local
Temperatures)
Point Kinetics
(Feedback
from Core
average Temp.)
1.00000
1.00000
1.22856
1.64098
2.27518
3.26028
11.07470
7.30756
5.26561
3.65653
2.50223
1.72007
(2.07)
( -3.3)
(-13.2)
(-25.5)
(-41.0)
( -0.2)
( -6.3)
( -2.5)
(
(
1.8)
6.8)
( 11.2)
1.22831
1.64442
2.30771
3.43494
27.66200
13.32640
9.32945
6.01209
3.83647
2.50007
(2.35)
Quasistatic
Quasi-static
(10 Flux Updates)
1.00000
( -3.3)
(-13.0)
(-24.5)
(-37.9)
(149.3)
( 70.8)
( 72.8)
( 67.3)
( 63.8)
(61.6)
1.22840
1.80472
2.91833
5.27298
10.98920
7.75787
5.45006
3.76891
2.45800
1.72207
(9.27)
18 Flux
Updates)
1.00000
-3.2)
-4.6)
-4.5)
-4.6)
-1.0)
-0.5)
0.9)
4.9)
4.9)
11.3)
1.27345
1.88758
3.08216
5.60374
10.70220
7.73877
5.44461
3.72162
2.46688
1.63752
(14.21)
( 0.3)
(-0.2)
( 0.8)
( 1.4)
(-3.6)
(-0.8)
(0.8)
( 3.6)
( 5.3)
(5.9)
-38-
accurate.
However, again the computation time is longer
(a factor of 2).
With only 10 flux updates the computation
time is reduced to 9.2 seconds.
However the maximum error
increases up to 11.3%.
The tentative overall conclusions from these numerical test cases are:
i)
It is essential to use a precomputed curve of rod
worth vs. position in performing point kinetics studies.
ii)
It is preferable to use local temperature coeffi-
cients and temperature changes to compute feedback reactivities rather than average values.
iii)
The quasi-static method can yield acceptably accu-
rate results, but the cost may be higher than that of running
the full 3D calculations.
A much more thorough investigation will be required
before any firm conclusion about the overall validity of the
point kinetics model can be drawn.
However, these first,
simple tests indicate that, for the rod withdrawal and coldwater accident, it is a very poor approximation.
llisM
.
E
1
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